LMFDB - Embedded newform 7935.2.a.j.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7935,2,Mod(1,7935)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7935 = 3 \cdot 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7935.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,1,-1,1,1,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.3612940039$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7935.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -4.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -4.00000 q^{22} -3.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -10.0000 q^{29} +1.00000 q^{30} -8.00000 q^{31} +5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -4.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} +2.00000 q^{41} -4.00000 q^{42} +8.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +12.0000 q^{56} +4.00000 q^{57} -10.0000 q^{58} -1.00000 q^{60} -6.00000 q^{61} -8.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +6.00000 q^{65} -4.00000 q^{66} -8.00000 q^{67} -2.00000 q^{68} -4.00000 q^{70} -4.00000 q^{71} -3.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +16.0000 q^{77} +6.00000 q^{78} -16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} +4.00000 q^{84} +2.00000 q^{85} +8.00000 q^{86} -10.0000 q^{87} +12.0000 q^{88} +10.0000 q^{89} +1.00000 q^{90} -24.0000 q^{91} -8.00000 q^{93} +4.00000 q^{95} +5.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	−4.00000	−1.06904
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	−4.00000	−0.872872
$22$	−4.00000	−0.852803
$23$	0	0
$23$	0	0
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	1.00000	0.182574
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.00000	0.883883
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	−4.00000	−0.676123
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	4.00000	0.648886
$39$	6.00000	0.960769
$40$	−3.00000	−0.474342
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−4.00000	−0.617213
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	−6.00000	−0.832050
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	−4.00000	−0.539360
$56$	12.0000	1.60357
$57$	4.00000	0.529813
$58$	−10.0000	−1.31306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−1.00000	−0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−8.00000	−1.01600
$63$	−4.00000	−0.503953
$64$	7.00000	0.875000
$65$	6.00000	0.744208
$66$	−4.00000	−0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−4.00000	−0.478091
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−3.00000	−0.353553
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	16.0000	1.82337
$78$	6.00000	0.679366
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	4.00000	0.436436
$85$	2.00000	0.216930
$86$	8.00000	0.862662
$87$	−10.0000	−1.07211
$88$	12.0000	1.27920
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	1.00000	0.105409
$91$	−24.0000	−2.51588
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	4.00000	0.410391
$96$	5.00000	0.510310
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	9.00000	0.909137
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	2.00000	0.198030
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	−18.0000	−1.76505
$105$	−4.00000	−0.390360
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−4.00000	−0.381385
$111$	−2.00000	−0.189832
$112$	4.00000	0.377964
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	10.0000	0.928477
$117$	6.00000	0.554700
$118$	0	0
$119$	−8.00000	−0.733359
$120$	−3.00000	−0.273861
$121$	5.00000	0.454545
$122$	−6.00000	−0.543214
$123$	2.00000	0.180334
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	−4.00000	−0.356348
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−3.00000	−0.265165
$129$	8.00000	0.704361
$130$	6.00000	0.526235
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	4.00000	0.348155
$133$	−16.0000	−1.38738
$134$	−8.00000	−0.691095
$135$	1.00000	0.0860663
$136$	−6.00000	−0.514496
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−4.00000	−0.335673
$143$	−24.0000	−2.00698
$144$	−1.00000	−0.0833333
$145$	−10.0000	−0.830455
$146$	10.0000	0.827606
$147$	9.00000	0.742307
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−12.0000	−0.973329
$153$	2.00000	0.161690
$154$	16.0000	1.28932
$155$	−8.00000	−0.642575
$156$	−6.00000	−0.480384
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−16.0000	−1.27289
$159$	6.00000	0.475831
$160$	5.00000	0.395285
$161$	0	0
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−2.00000	−0.156174
$165$	−4.00000	−0.311400
$166$	12.0000	0.931381
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	12.0000	0.925820
$169$	23.0000	1.76923
$170$	2.00000	0.153393
$171$	4.00000	0.305888
$172$	−8.00000	−0.609994
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	−10.0000	−0.758098
$175$	−4.00000	−0.302372
$176$	4.00000	0.301511
$177$	0	0
$178$	10.0000	0.749532
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−1.00000	−0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−24.0000	−1.77900
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−2.00000	−0.147043
$186$	−8.00000	−0.586588
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−4.00000	−0.290957
$190$	4.00000	0.290191
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	7.00000	0.505181
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	10.0000	0.717958
$195$	6.00000	0.429669
$196$	−9.00000	−0.642857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−4.00000	−0.284268
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−3.00000	−0.212132
$201$	−8.00000	−0.564276
$202$	14.0000	0.985037
$203$	40.0000	2.80745
$204$	−2.00000	−0.140028
$205$	2.00000	0.139686
$206$	12.0000	0.836080
$207$	0	0
$208$	−6.00000	−0.416025
$209$	−16.0000	−1.10674
$210$	−4.00000	−0.276026
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	−4.00000	−0.274075
$214$	12.0000	0.820303
$215$	8.00000	0.545595
$216$	−3.00000	−0.204124
$217$	32.0000	2.17230
$218$	10.0000	0.677285
$219$	10.0000	0.675737
$220$	4.00000	0.269680
$221$	12.0000	0.807207
$222$	−2.00000	−0.134231
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	−20.0000	−1.33631
$225$	1.00000	0.0666667
$226$	2.00000	0.133038
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−4.00000	−0.264906
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	30.0000	1.96960
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	−8.00000	−0.518563
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	−1.00000	−0.0645497
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	9.00000	0.574989
$246$	2.00000	0.127515
$247$	24.0000	1.52708
$248$	24.0000	1.52400
$249$	12.0000	0.760469
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	0	0
$255$	2.00000	0.125245
$256$	−17.0000	−1.06250
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	8.00000	0.498058
$259$	8.00000	0.497096
$260$	−6.00000	−0.372104
$261$	−10.0000	−0.618984
$262$	8.00000	0.494242
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	12.0000	0.738549
$265$	6.00000	0.368577
$266$	−16.0000	−0.981023
$267$	10.0000	0.611990
$268$	8.00000	0.488678
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	1.00000	0.0608581
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−2.00000	−0.121268
$273$	−24.0000	−1.45255
$274$	10.0000	0.604122
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	−8.00000	−0.478947
$280$	12.0000	0.717137
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	4.00000	0.237356
$285$	4.00000	0.236940
$286$	−24.0000	−1.41915
$287$	−8.00000	−0.472225
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	−10.0000	−0.587220
$291$	10.0000	0.586210
$292$	−10.0000	−0.585206
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	6.00000	0.348743
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−32.0000	−1.84445
$302$	−16.0000	−0.920697
$303$	14.0000	0.804279
$304$	−4.00000	−0.229416
$305$	−6.00000	−0.343559
$306$	2.00000	0.114332
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	−16.0000	−0.911685
$309$	12.0000	0.682656
$310$	−8.00000	−0.454369
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−18.0000	−1.01905
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	14.0000	0.790066
$315$	−4.00000	−0.225374
$316$	16.0000	0.900070
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	6.00000	0.336463
$319$	40.0000	2.23957
$320$	7.00000	0.391312
$321$	12.0000	0.669775
$322$	0	0
$323$	8.00000	0.445132
$324$	−1.00000	−0.0555556
$325$	6.00000	0.332820
$326$	20.0000	1.10770
$327$	10.0000	0.553001
$328$	−6.00000	−0.331295
$329$	0	0
$330$	−4.00000	−0.220193
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−12.0000	−0.658586
$333$	−2.00000	−0.109599
$334$	8.00000	0.437741
$335$	−8.00000	−0.437087
$336$	4.00000	0.218218
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	23.0000	1.25104
$339$	2.00000	0.108625
$340$	−2.00000	−0.108465
$341$	32.0000	1.73290
$342$	4.00000	0.216295
$343$	−8.00000	−0.431959
$344$	−24.0000	−1.29399
$345$	0	0
$346$	2.00000	0.107521
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	10.0000	0.536056
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−4.00000	−0.213809
$351$	6.00000	0.320256
$352$	−20.0000	−1.06600
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	−10.0000	−0.529999
$357$	−8.00000	−0.423405
$358$	24.0000	1.26844
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	−3.00000	−0.158114
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	5.00000	0.262432
$364$	24.0000	1.25794
$365$	10.0000	0.523424
$366$	−6.00000	−0.313625
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	−2.00000	−0.103975
$371$	−24.0000	−1.24602
$372$	8.00000	0.414781
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−8.00000	−0.413670
$375$	1.00000	0.0516398
$376$	0	0
$377$	−60.0000	−3.09016
$378$	−4.00000	−0.205738
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	−24.0000	−1.22795
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	−3.00000	−0.153093
$385$	16.0000	0.815436
$386$	−6.00000	−0.305392
$387$	8.00000	0.406663
$388$	−10.0000	−0.507673
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	6.00000	0.303822
$391$	0	0
$392$	−27.0000	−1.36371
$393$	8.00000	0.403547
$394$	−6.00000	−0.302276
$395$	−16.0000	−0.805047
$396$	4.00000	0.201008
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	−24.0000	−1.20301
$399$	−16.0000	−0.801002
$400$	−1.00000	−0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−8.00000	−0.399004
$403$	−48.0000	−2.39105
$404$	−14.0000	−0.696526
$405$	1.00000	0.0496904
$406$	40.0000	1.98517
$407$	8.00000	0.396545
$408$	−6.00000	−0.297044
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	2.00000	0.0987730
$411$	10.0000	0.493264
$412$	−12.0000	−0.591198
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	30.0000	1.47087
$417$	4.00000	0.195881
$418$	−16.0000	−0.782586
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	4.00000	0.195180
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	−18.0000	−0.874157
$425$	2.00000	0.0970143
$426$	−4.00000	−0.193801
$427$	24.0000	1.16144
$428$	−12.0000	−0.580042
$429$	−24.0000	−1.15873
$430$	8.00000	0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	32.0000	1.53605
$435$	−10.0000	−0.479463
$436$	−10.0000	−0.478913
$437$	0	0
$438$	10.0000	0.477818
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	12.0000	0.572078
$441$	9.00000	0.428571
$442$	12.0000	0.570782
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	2.00000	0.0949158
$445$	10.0000	0.474045
$446$	24.0000	1.13643
$447$	6.00000	0.283790
$448$	−28.0000	−1.32288
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	1.00000	0.0471405
$451$	−8.00000	−0.376705
$452$	−2.00000	−0.0940721
$453$	−16.0000	−0.751746
$454$	12.0000	0.563188
$455$	−24.0000	−1.12514
$456$	−12.0000	−0.561951
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−6.00000	−0.280362
$459$	2.00000	0.0933520
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	16.0000	0.744387
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	10.0000	0.464238
$465$	−8.00000	−0.370991
$466$	−2.00000	−0.0926482
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	−6.00000	−0.277350
$469$	32.0000	1.47762
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	−32.0000	−1.47136
$474$	−16.0000	−0.734904
$475$	4.00000	0.183533
$476$	8.00000	0.366679
$477$	6.00000	0.274721
$478$	20.0000	0.914779
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	5.00000	0.228218
$481$	−12.0000	−0.547153
$482$	14.0000	0.637683
$483$	0	0
$484$	−5.00000	−0.227273
$485$	10.0000	0.454077
$486$	1.00000	0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	18.0000	0.814822
$489$	20.0000	0.904431
$490$	9.00000	0.406579
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	−2.00000	−0.0901670
$493$	−20.0000	−0.900755
$494$	24.0000	1.07981
$495$	−4.00000	−0.179787
$496$	8.00000	0.359211
$497$	16.0000	0.717698
$498$	12.0000	0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	8.00000	0.357414
$502$	12.0000	0.535586
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	12.0000	0.534522
$505$	14.0000	0.622992
$506$	0	0
$507$	23.0000	1.02147
$508$	0	0
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	2.00000	0.0885615
$511$	−40.0000	−1.76950
$512$	−11.0000	−0.486136
$513$	4.00000	0.176604
$514$	30.0000	1.32324
$515$	12.0000	0.528783
$516$	−8.00000	−0.352180
$517$	0	0
$518$	8.00000	0.351500
$519$	2.00000	0.0877903
$520$	−18.0000	−0.789352
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−10.0000	−0.437688
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	−8.00000	−0.349482
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−16.0000	−0.696971
$528$	4.00000	0.174078
$529$	0	0
$530$	6.00000	0.260623
$531$	0	0
$532$	16.0000	0.693688
$533$	12.0000	0.519778
$534$	10.0000	0.432742
$535$	12.0000	0.518805
$536$	24.0000	1.03664
$537$	24.0000	1.03568
$538$	−2.00000	−0.0862261
$539$	−36.0000	−1.55063
$540$	−1.00000	−0.0430331
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	16.0000	0.687259
$543$	2.00000	0.0858282
$544$	10.0000	0.428746
$545$	10.0000	0.428353
$546$	−24.0000	−1.02711
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−10.0000	−0.427179
$549$	−6.00000	−0.256074
$550$	−4.00000	−0.170561
$551$	−40.0000	−1.70406
$552$	0	0
$553$	64.0000	2.72156
$554$	−10.0000	−0.424859
$555$	−2.00000	−0.0848953
$556$	−4.00000	−0.169638
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	−8.00000	−0.338667
$559$	48.0000	2.03018
$560$	4.00000	0.169031
$561$	−8.00000	−0.337760
$562$	−6.00000	−0.253095
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	24.0000	1.00880
$567$	−4.00000	−0.167984
$568$	12.0000	0.503509
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	4.00000	0.167542
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	24.0000	1.00349
$573$	−24.0000	−1.00261
$574$	−8.00000	−0.333914
$575$	0	0
$576$	7.00000	0.291667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	−13.0000	−0.540729
$579$	−6.00000	−0.249351
$580$	10.0000	0.415227
$581$	−48.0000	−1.99138
$582$	10.0000	0.414513
$583$	−24.0000	−0.993978
$584$	−30.0000	−1.24141
$585$	6.00000	0.248069
$586$	−26.0000	−1.07405
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	−9.00000	−0.371154
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−6.00000	−0.246807
$592$	2.00000	0.0821995
$593$	46.0000	1.88899	0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000	1.88899	0.944497	+	0.328521i	$0.106550\pi$
$594$	−4.00000	−0.164122
$595$	−8.00000	−0.327968
$596$	−6.00000	−0.245770
$597$	−24.0000	−0.982255
$598$	0	0
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	−3.00000	−0.122474
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−32.0000	−1.30422
$603$	−8.00000	−0.325785
$604$	16.0000	0.651031
$605$	5.00000	0.203279
$606$	14.0000	0.568711
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	20.0000	0.811107
$609$	40.0000	1.62088
$610$	−6.00000	−0.242933
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−28.0000	−1.12999
$615$	2.00000	0.0806478
$616$	−48.0000	−1.93398
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	12.0000	0.482711
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	12.0000	0.481156
$623$	−40.0000	−1.60257
$624$	−6.00000	−0.240192
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	−16.0000	−0.638978
$628$	−14.0000	−0.558661
$629$	−4.00000	−0.159490
$630$	−4.00000	−0.159364
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	48.0000	1.90934
$633$	−4.00000	−0.158986
$634$	−6.00000	−0.238290
$635$	0	0
$636$	−6.00000	−0.237915
$637$	54.0000	2.13956
$638$	40.0000	1.58362
$639$	−4.00000	−0.158238
$640$	−3.00000	−0.118585
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	12.0000	0.473602
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	8.00000	0.314756
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	6.00000	0.235339
$651$	32.0000	1.25418
$652$	−20.0000	−0.783260
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	10.0000	0.391031
$655$	8.00000	0.312586
$656$	−2.00000	−0.0780869
$657$	10.0000	0.390137
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	4.00000	0.155700
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	12.0000	0.466393
$663$	12.0000	0.466041
$664$	−36.0000	−1.39707
$665$	−16.0000	−0.620453
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−8.00000	−0.309529
$669$	24.0000	0.927894
$670$	−8.00000	−0.309067
$671$	24.0000	0.926510
$672$	−20.0000	−0.771517
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	26.0000	1.00148
$675$	1.00000	0.0384900
$676$	−23.0000	−0.884615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	2.00000	0.0768095
$679$	−40.0000	−1.53506
$680$	−6.00000	−0.230089
$681$	12.0000	0.459841
$682$	32.0000	1.22534
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	−4.00000	−0.152944
$685$	10.0000	0.382080
$686$	−8.00000	−0.305441
$687$	−6.00000	−0.228914
$688$	−8.00000	−0.304997
$689$	36.0000	1.37149
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−2.00000	−0.0760286
$693$	16.0000	0.607790
$694$	−12.0000	−0.455514
$695$	4.00000	0.151729
$696$	30.0000	1.13715
$697$	4.00000	0.151511
$698$	14.0000	0.529908
$699$	−2.00000	−0.0756469
$700$	4.00000	0.151186
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	6.00000	0.226455
$703$	−8.00000	−0.301726
$704$	−28.0000	−1.05529
$705$	0	0
$706$	6.00000	0.225813
$707$	−56.0000	−2.10610
$708$	0	0
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	−4.00000	−0.150117
$711$	−16.0000	−0.600047
$712$	−30.0000	−1.12430
$713$	0	0
$714$	−8.00000	−0.299392
$715$	−24.0000	−0.897549
$716$	−24.0000	−0.896922
$717$	20.0000	0.746914
$718$	−8.00000	−0.298557
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	−1.00000	−0.0372678
$721$	−48.0000	−1.78761
$722$	−3.00000	−0.111648
$723$	14.0000	0.520666
$724$	−2.00000	−0.0743294
$725$	−10.0000	−0.371391
$726$	5.00000	0.185567
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	72.0000	2.66850
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	16.0000	0.591781
$732$	6.00000	0.221766
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	−28.0000	−1.03350
$735$	9.00000	0.331970
$736$	0	0
$737$	32.0000	1.17874
$738$	2.00000	0.0736210
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	2.00000	0.0735215
$741$	24.0000	0.881662
$742$	−24.0000	−0.881068
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	24.0000	0.879883
$745$	6.00000	0.219823
$746$	−10.0000	−0.366126
$747$	12.0000	0.439057
$748$	8.00000	0.292509
$749$	−48.0000	−1.75388
$750$	1.00000	0.0365148
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	−60.0000	−2.18507
$755$	−16.0000	−0.582300
$756$	4.00000	0.145479
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	−12.0000	−0.435286
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−40.0000	−1.44810
$764$	24.0000	0.868290
$765$	2.00000	0.0723102
$766$	−32.0000	−1.15621
$767$	0	0
$768$	−17.0000	−0.613435
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	16.0000	0.576600
$771$	30.0000	1.08042
$772$	6.00000	0.215945
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	8.00000	0.287554
$775$	−8.00000	−0.287368
$776$	−30.0000	−1.07694
$777$	8.00000	0.286998
$778$	14.0000	0.501924
$779$	8.00000	0.286630
$780$	−6.00000	−0.214834
$781$	16.0000	0.572525
$782$	0	0
$783$	−10.0000	−0.357371
$784$	−9.00000	−0.321429
$785$	14.0000	0.499681
$786$	8.00000	0.285351
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−16.0000	−0.569254
$791$	−8.00000	−0.284447
$792$	12.0000	0.426401
$793$	−36.0000	−1.27840
$794$	30.0000	1.06466
$795$	6.00000	0.212798
$796$	24.0000	0.850657
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	−16.0000	−0.566394
$799$	0	0
$800$	5.00000	0.176777
$801$	10.0000	0.353333
$802$	10.0000	0.353112
$803$	−40.0000	−1.41157
$804$	8.00000	0.282138
$805$	0	0
$806$	−48.0000	−1.69073
$807$	−2.00000	−0.0704033
$808$	−42.0000	−1.47755
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	1.00000	0.0351364
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	−40.0000	−1.40372
$813$	16.0000	0.561144
$814$	8.00000	0.280400
$815$	20.0000	0.700569
$816$	−2.00000	−0.0700140
$817$	32.0000	1.11954
$818$	10.0000	0.349642
$819$	−24.0000	−0.838628
$820$	−2.00000	−0.0698430
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	10.0000	0.348790
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	−36.0000	−1.25412
$825$	−4.00000	−0.139262
$826$	0	0
$827$	52.0000	1.80822	0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000	1.80822	0.904109	+	0.427303i	$0.140536\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	12.0000	0.416526
$831$	−10.0000	−0.346896
$832$	42.0000	1.45609
$833$	18.0000	0.623663
$834$	4.00000	0.138509
$835$	8.00000	0.276851
$836$	16.0000	0.553372
$837$	−8.00000	−0.276520
$838$	28.0000	0.967244
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	12.0000	0.414039
$841$	71.0000	2.44828
$842$	10.0000	0.344623
$843$	−6.00000	−0.206651
$844$	4.00000	0.137686
$845$	23.0000	0.791224
$846$	0	0
$847$	−20.0000	−0.687208
$848$	−6.00000	−0.206041
$849$	24.0000	0.823678
$850$	2.00000	0.0685994
$851$	0	0
$852$	4.00000	0.137038
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	24.0000	0.821263
$855$	4.00000	0.136797
$856$	−36.0000	−1.23045
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	−24.0000	−0.819346
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	−8.00000	−0.272798
$861$	−8.00000	−0.272639
$862$	0	0
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	5.00000	0.170103
$865$	2.00000	0.0680020
$866$	−14.0000	−0.475739
$867$	−13.0000	−0.441503
$868$	−32.0000	−1.08615
$869$	64.0000	2.17105
$870$	−10.0000	−0.339032
$871$	−48.0000	−1.62642
$872$	−30.0000	−1.01593
$873$	10.0000	0.338449
$874$	0	0
$875$	−4.00000	−0.135225
$876$	−10.0000	−0.337869
$877$	54.0000	1.82345	0.911725	−	0.410801i	$-0.134751\pi$
$877$	54.0000	1.82345	0.911725	+	0.410801i	$0.134751\pi$
$878$	−16.0000	−0.539974
$879$	−26.0000	−0.876958
$880$	4.00000	0.134840
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	9.00000	0.303046
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−12.0000	−0.403148
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	10.0000	0.335201
$891$	−4.00000	−0.134005
$892$	−24.0000	−0.803579
$893$	0	0
$894$	6.00000	0.200670
$895$	24.0000	0.802232
$896$	12.0000	0.400892
$897$	0	0
$898$	−14.0000	−0.467186
$899$	80.0000	2.66815
$900$	−1.00000	−0.0333333
$901$	12.0000	0.399778
$902$	−8.00000	−0.266371
$903$	−32.0000	−1.06489
$904$	−6.00000	−0.199557
$905$	2.00000	0.0664822
$906$	−16.0000	−0.531564
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	−12.0000	−0.398234
$909$	14.0000	0.464351
$910$	−24.0000	−0.795592
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−4.00000	−0.132453
$913$	−48.0000	−1.58857
$914$	18.0000	0.595387
$915$	−6.00000	−0.198354
$916$	6.00000	0.198246
$917$	−32.0000	−1.05673
$918$	2.00000	0.0660098
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	−34.0000	−1.11973
$923$	−24.0000	−0.789970
$924$	−16.0000	−0.526361
$925$	−2.00000	−0.0657596
$926$	−32.0000	−1.05159
$927$	12.0000	0.394132
$928$	−50.0000	−1.64133
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−8.00000	−0.262330
$931$	36.0000	1.17985
$932$	2.00000	0.0655122
$933$	12.0000	0.392862
$934$	−20.0000	−0.654420
$935$	−8.00000	−0.261628
$936$	−18.0000	−0.588348
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	32.0000	1.04484
$939$	−14.0000	−0.456873
$940$	0	0
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	14.0000	0.456145
$943$	0	0
$944$	0	0
$945$	−4.00000	−0.130120
$946$	−32.0000	−1.04041
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	16.0000	0.519656
$949$	60.0000	1.94768
$950$	4.00000	0.129777
$951$	−6.00000	−0.194563
$952$	24.0000	0.777844
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	6.00000	0.194257
$955$	−24.0000	−0.776622
$956$	−20.0000	−0.646846
$957$	40.0000	1.29302
$958$	−40.0000	−1.29234
$959$	−40.0000	−1.29167
$960$	7.00000	0.225924
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	12.0000	0.386695
$964$	−14.0000	−0.450910
$965$	−6.00000	−0.193147
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−15.0000	−0.482118
$969$	8.00000	0.256997
$970$	10.0000	0.321081
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	−1.00000	−0.0320750
$973$	−16.0000	−0.512936
$974$	24.0000	0.769010
$975$	6.00000	0.192154
$976$	6.00000	0.192055
$977$	50.0000	1.59964	0.799821	−	0.600239i	$-0.204928\pi$
$977$	50.0000	1.59964	0.799821	+	0.600239i	$0.204928\pi$
$978$	20.0000	0.639529
$979$	−40.0000	−1.27841
$980$	−9.00000	−0.287494
$981$	10.0000	0.319275
$982$	16.0000	0.510581
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	−6.00000	−0.191273
$985$	−6.00000	−0.191176
$986$	−20.0000	−0.636930
$987$	0	0
$988$	−24.0000	−0.763542
$989$	0	0
$990$	−4.00000	−0.127128
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−40.0000	−1.27000
$993$	12.0000	0.380808
$994$	16.0000	0.507489
$995$	−24.0000	−0.760851
$996$	−12.0000	−0.380235
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	4.00000	0.126618
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7935.2.a.j.1.1		1
23.22	odd	2		345.2.a.e.1.1	✓	1
69.68	even	2		1035.2.a.c.1.1		1
92.91	even	2		5520.2.a.a.1.1		1
115.22	even	4		1725.2.b.f.1174.2		2
115.68	even	4		1725.2.b.f.1174.1		2
115.114	odd	2		1725.2.a.b.1.1		1
345.344	even	2		5175.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.e.1.1	✓	1	23.22	odd	2
1035.2.a.c.1.1		1	69.68	even	2
1725.2.a.b.1.1		1	115.114	odd	2
1725.2.b.f.1174.1		2	115.68	even	4
1725.2.b.f.1174.2		2	115.22	even	4
5175.2.a.q.1.1		1	345.344	even	2
5520.2.a.a.1.1		1	92.91	even	2
7935.2.a.j.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7935.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more